Theorem isosolem 5725

Description: Lemma for isoso 5726. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Assertion

Ref	Expression
isosolem	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Proof of Theorem isosolem

Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isopolem 5723	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
2		df-3an 964	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴))
3		isof1o 5708	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1of 5367	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
5		ffvelrn 5553	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ 𝐵)
6	5	ex 114	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑎 ∈ 𝐴 → (𝐻‘𝑎) ∈ 𝐵))
7		ffvelrn 5553	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑏 ∈ 𝐴) → (𝐻‘𝑏) ∈ 𝐵)
8	7	ex 114	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑏 ∈ 𝐴 → (𝐻‘𝑏) ∈ 𝐵))
9		ffvelrn 5553	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑐 ∈ 𝐴) → (𝐻‘𝑐) ∈ 𝐵)
10	9	ex 114	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑐 ∈ 𝐴 → (𝐻‘𝑐) ∈ 𝐵))
11	6, 8, 10	3anim123d 1297	. . . . . . . . . . 11 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵)))
12	3, 4, 11	3syl 17	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵)))
13	12	imp 123	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵))
14		breq1 3932	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐻‘𝑎) → (𝑥𝑆𝑦 ↔ (𝐻‘𝑎)𝑆𝑦))
15		breq1 3932	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐻‘𝑎) → (𝑥𝑆𝑧 ↔ (𝐻‘𝑎)𝑆𝑧))
16	15	orbi1d 780	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐻‘𝑎) → ((𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦) ↔ ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦)))
17	14, 16	imbi12d 233	. . . . . . . . . 10 ⊢ (𝑥 = (𝐻‘𝑎) → ((𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) ↔ ((𝐻‘𝑎)𝑆𝑦 → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦))))
18		breq2 3933	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑦 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
19		breq2 3933	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐻‘𝑏) → (𝑧𝑆𝑦 ↔ 𝑧𝑆(𝐻‘𝑏)))
20	19	orbi2d 779	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻‘𝑏) → (((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦) ↔ ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏))))
21	18, 20	imbi12d 233	. . . . . . . . . 10 ⊢ (𝑦 = (𝐻‘𝑏) → (((𝐻‘𝑎)𝑆𝑦 → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆𝑦)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏)))))
22		breq2 3933	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑐) → ((𝐻‘𝑎)𝑆𝑧 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
23		breq1 3932	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐻‘𝑐) → (𝑧𝑆(𝐻‘𝑏) ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
24	22, 23	orbi12d 782	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐻‘𝑐) → (((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏))))
25	24	imbi2d 229	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻‘𝑐) → (((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆𝑧 ∨ 𝑧𝑆(𝐻‘𝑏))) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
26	17, 21, 25	rspc3v 2805	. . . . . . . . 9 ⊢ (((𝐻‘𝑎) ∈ 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑐) ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
27	13, 26	syl 14	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
28		isorel 5709	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
29	28	3adantr3 1142	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
30		isorel 5709	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
31	30	3adantr2 1141	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑐)))
32		isorel 5709	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
33	32	ancom2s 555	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
34	33	3adantr1 1140	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))
35	31, 34	orbi12d 782	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ((𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏))))
36	29, 35	imbi12d 233	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ((𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)) ↔ ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑐) ∨ (𝐻‘𝑐)𝑆(𝐻‘𝑏)))))
37	27, 36	sylibrd 168	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
38	2, 37	sylan2br 286	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
39	38	anassrs 397	. . . . 5 ⊢ (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) ∧ 𝑐 ∈ 𝐴) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
40	39	ralrimdva 2512	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
41	40	ralrimdvva 2517	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦)) → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
42	1, 41	anim12d 333	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)))))
43		df-iso 4219	. 2 ⊢ (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑧𝑆𝑦))))
44		df-iso 4219	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
45	42, 43, 44	3imtr4g 204	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416   class class class wbr 3929   Po wpo 4216   Or wor 4217  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123   Isom wiso 5124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-f1o 5130  df-fv 5131  df-isom 5132

This theorem is referenced by:  isoso  5726